Magnetic field-induced ferroelectricity in S = 1/2 kagome staircase compound PbCu3TeO7

Ni3V2O8 is an archetypical multiferroic material with a kagome staircase structure of S = 1 (Ni2+) spins whose complex interplay between spin ordering and ferroelectricity has been studied for more than a decade. Here, we report a new kagome staircase compound PbCu3TeO7 with Cu2+ (S = 1/2) spins that exhibits two Néel temperatures at TN1 = 36 K and TN2 = 24 K, and a magnetic field (H)-induced electric polarization (P) below TN2. Pyroelectric and magnetoelectric current measurements in magnetic fields up to 60 T reveal that for H||c of ~ 8.3 T, a spin-flop transition induces a transverse P||a with a magnitude of 15 µC/m2 below TN2. Furthermore, for a parallel configuration with P//H//a, two spin-flop transitions occur, the first at ~ 16 T with P//a of 14 µC/m2, and the second at ~ 38 T, where P disappears. Monte Carlo simulations based on 12 major exchange interactions uncover that a sinusoidal amplitude modulation of the spins occurs along the b-axis below TN1 and an incommensurate, proper screw-type spin order occurs in the ac-plane below TN2. The simulation results show that P//a under H//c stems from a spin-flop transition facilitating an ab-plane-type spiral order, while the two successive spin-flop transitions for H//a result in spiral spin orders in the ab- and bc-planes. Based on the experimental and theoretical results, we establish field-induced magnetic and electric phase diagrams for the two H directions, demonstrating that the distorted kagome staircase structure with competing intra–interlayer interactions and lifted frustration creates a plethora of different noncollinear spin textures of S = 1/2 spins that in turn induce electric polarization. In multiferroic materials, the ferroelectric polarization can be induced by the spin order; now a Cu-based material with a geometrically frustrated (kagome) lattice joins the list of oxides in which this effect has been observed. In an ideal material the frustration would give rise to degenerate ground states such as the quantum-spin liquid, but the presence of lattice defects lifts the spin frustration, allowing the formation of a long-range spin order. Kee Hoon Kim from Seoul National University and colleagues observed magnetic-field-induced ferroelectricity in a material containing Cu2+ ions with spin 1/2 that has a kagome spin structure. They established the phase diagram of the material for different directions of the magnetic field and attributed the origin of the polarization to the spin structure by means of Monte Carlo simulations.


INTRODUCTION
Research activities on multiferroics and magnetoelectric (ME) materials have resumed since early 2000s because of the fundamental interests in the physics of strong coupling between spin and lattice degrees of freedom as well as the application potential for numerous low-power electronic devices. [1][2][3][4][5] In particular, magnetic ferroelectrics, in which ferroelectric polarization (P) is induced by spin order, have drawn lots of attention due to the intriguing complexity of the spin textures and magnetoelectric coupling mechanisms, as well as the ability to control 100% of the electric polarization with magnetic ordering. In magnetic ferroelectrics, one route to creating ferroelectricity is via competing magnetic interactions that induce complex nontrivial spin orders. These complex spin orders in turn break spatial inversion symmetry of the combined spin-lattice system and thus induce P. 4,5 For example, ferroelectricity observed in quasi-onedimensional spin chain compounds MnWO 4 , 6 Ca 3 Co 1.04 Mn 0.96 O 6 , 7 and Lu 2 CoMnO 6 8 can be explained by such complex spin orders induced by the competing nearest and next-nearest neighbor exchange interactions.
Another effective way to realize complex spin orders is to make use of "lifted frustration". In geometrically frustrated lattice such as triangular, kagome, or spinel structures, the frustration is so complete that no single ordered ground state can occur in theory, giving rise to classical or quantum-spin liquids. However, in real systems, the perfect spin frustration is often lifted by symmetryreducing lattice distortions that eventually lead to spontaneous spin ordering at finite temperatures. These lattice distortions can directly cause ferroelectricity, or the resulting spin orderings are also sufficiently complex as to induce electric polarization, as occurs, e.g., in Ni 3 V 2 O 8 , [9][10][11][12][13][14] CuFeO 2 ,15,16 CoCr 2 O 4 17 , and AgCrO 2 . 18 Among geometrically frustrated lattices, 6-8 the kagome structure has received particular attention since it is a prime candidate for creating quantum-spin-liquid states. 19 Numerous studies 20,21 have mainly focused on the quest for a possible quantum-spinliquid ground state in the ideal kagome lattices, whereas multiferroic properties have not been extensively studied. Ni 3 V 2 O 8 , however, is a rare example of exhibiting multiferroic behavior in a distorted kagome structure. Here, Ni 2+ (S = 1) spins organize into a buckled form of the kagome structure, called as the kagome staircase structure. Extensive studies on Ni 3 V 2 O 8 have now made significant progresses in understanding its intriguing ME coupling and complex phase diagram. [9][10][11][12][13][14] Upon being cooled, Ni 3 V 2 O 8 exhibits four magnetic phases below 9.3 K; high-and lowtemperature incommensurate (HTI and LTI) and two commensurate ones. Among those, ferroelectricity has been found only in the LTI phase, in which neutron scattering studies 9,12 have revealed stabilization of a spin cycloid structure in the ab-plane. Therefore, the origin of spontaneous P is now understood as coming from the so-called inverse Dzyaloshinskii-Moriya (IDM) mechanism. [22][23][24] In terms of multiferroics/ME properties, the S = 1/2 system containing Cu 2+ ions has drawn continual interest because strong quantum fluctuation effects expected from the smallest possible spin value may show up in the magnetic and multiferroic properties. 25,26 Moreover, some of the Cu 2+ -containing compounds have high spin ordering temperatures because of their strong superexchange interaction arising from Cu-O-Cu bonding. 27 36 and KCu 3 As 2 O 7 (OD) 3 . 37,38 Most of these compounds listed in refs. 25-38 share a common feature of CuO 2 ribbons with edge-sharing CuO 4 square planes. Therefore, the nearest-neighbor spin interaction through the Cu-O-Cu path can compete with the next-nearest neighbor interaction through the Cu-O-O-Cu path, resulting in a cycloidal spin order. Spontaneous P can then be induced by the IDM mechanism. [22][23][24] Moreover, recently known multiferroic compound KCu 3 As 2 O 7 (OD) 3 has a monoclinically distorted kagome structure, which allows long-range spin order with a cycloidal configuration. 37,38 However, to our knowledge, no ME or multiferroic behaviors have been found to date in the staircase kagome spin structure with S = 1/2 Cu 2+ ions, which could open a new route to find exotic magnetic and multiferroic phenomena due to its unique opportunity of having quantum fluctuation, geometric spin frustration, and competing nearest-and nextnearest spin interaction all in one spin network.
We have recently found that Cu 2+ (S = 1/2) ions in one of emergent copper tellurium oxides PbCu 3 TeO 7 , 39-41 form the kagome staircase structure. The previous studies of magnetization, 39 heat capacity, 39 and nuclear magnetic resonance (NMR) studies 40 have shown that PbCu 3 TeO 7 has two antiferromagnetic (AFM) transitions at T N1 = 36 K and T N2 = 24 K and a small change in the magnetic susceptibility at T N3 = 17 K. The Curie-Weiss temperature θ CW = −150 K obtained from the magnetic susceptibility indicated the presence of moderate spin frustration. In addition, strong interlayer interactions through the Cu-O-Te-O-Cu path were evident in the NMR measurement. 40 This observation indicates that we should expect not only highly frustrated intralayer spin interactions, but also competing nextnearest exchange interaction via the strong interlayer coupling. Therefore, PbCu 3 TeO 7 is a good candidate material for exhibiting the noncollinear spin order and magnetism-induced ferroelectricity.
In this article, we report the observation of magnetic-field (H)induced ferroelectricity in PbCu 3 TeO 7 . We find that below T N2 = 24 K, P a is generated for transverse µ 0 H c > 8.3 T and longitudinal 16 T < µ 0 H a < 38 T, due to spin-flop (SF) transitions. We establish the phase diagram of PbCu 3 TeO 7 for each H direction based on the electric and magnetic properties. Based on the exchange interactions obtained by the first-principles calculation, Monte Carlo (MC) simulations were performed to reveal spin configurations in each phase. Our calculations suggest that the P a generated by µ 0 H c > 8.3 T and 16 T < µ 0 H a < 38 T stems from a cycloid spin phase rotating in the ab-plane.

RESULTS
PbCu 3 TeO 7 crystallizes in an orthorhombic structure (Pnma space group), in which Cu atoms form the two-dimensional kagome structure in the bc-plane (Fig. 1a, b). The kagome planes are buckled in a zigzag form to constitute the staircase kagome structure. The buckled kagome planes are also stacked along the a-direction and separated by Pb and Te atoms with a small interlayer distance of~4 Å. The lattice parameters were obtained from the Rietveld refinement of powder XRD patterns; a = 10.484 Å, b = 6.347 Å, and c = 8.807 Å. 39 First-principles calculation based on the density functional theory (DFT) were performed to determine the values of the possible exchange interactions. Figure  1c summarizes the main intraplane exchange interactions labeled . c Possible intralayer exchange interactions denoted from J1 to J8 in the order of the distance. d Temperature-dependent magnetic susceptibility data along each field direction from J 1 to J 8 in the order of the distance between neighboring Cu pairs [Four additional interplane exchange interactions are not plotted here, but are summarized in Table 1]. Figure 1d plots the magnetic susceptibility χ(Τ) data measured along each crystallographic direction. χ exhibits significant anisotropy consistent with the previous report by Koteswararao et al. 39 For H//a, there is an abrupt decrease of χ below T N1 =~36 K, signaling an AFM ordering, termed AFM 1. For H//c, χ starts to decrease below T N2 =~24 K and eventually becomes lower than the χ value measured along the a-axis, implying stabilization of another AFM ordering, termed AFM 2. However, for H//b, χ remains high without showing any anomaly. All these observations indicate that the crystallographic b-direction is the spin-hard axis in the two AFM phases, while the a-direction is the spin-easy axis in the AFM 1 phase. Moreover, with the stabilization of the AFM 2 phase, the crystallographic c-direction also becomes the spin-easy axis. Besides, a small kink is observed at~17 K in the χ curve along H//a. It is also seen that the χ(Τ) curves along H//a and H//c cross each other at the anomaly at~17 K. Figure 2a-d present the electrical properties, i.e., the dielectric constant (ε a ), pyrocurrent I p , and the corresponding electric polarization (P a ) curves. All are measured along the a-direction in the H//c configuration (H c ). In the low-field region (Fig. 2a), the ε a curve exhibits a small kink in the vicinity of T N2 =~24 K. With increasing µ 0 H c , the kink feature grows and develops into a sharp peak; the peak intensity starts to increase above 6 T, shows an abrupt jump at µ 0 H c = 9 T, and becomes almost saturated for µ 0 H c ≥ 10 T (Fig. 2b). Finally, at the highest µ 0 H c = 14 T, ε a exhibits the sharpest single peak. Consistent with this, we observe a dip feature in the I p curves at µ 0 H c ≥~6 T (Fig. 2c). As the magnetic field increases beyond 8 T, the dip feature is progressively enhanced, exhibiting the most abrupt change between 8 and 9 T. Correspondingly, a finite P a starts to appear at µ 0 H c ≥ 6 T and exhibits an abrupt increase between 8 and 9 T. The resultant P a value at 2 K is summarized in Fig. 2e; it starts to appear at µ 0 H c ≥ 6 T, suddenly jumps between 8 and 9 T, and reaches the maximum value of 19.5 µC/m 2 at µ 0 H c = 14 T. The temperature, where P a starts to appear, almost coincides with T N2 = 24 K, implying that the AFM 2 phase at µ 0 H c ≥ 6 T is also ferroelectric. The sign of I p is reversed after negative electric poling (−E P ) (not shown here), supporting the existence of ferroelectricity at µ 0 H c ≥ 6 T. In addition, for magnetic fields up to 14 T, the dip feature in I p was only observed in this particular electric and magnetic field configuration, i.e., P//a and H//c.
To further understand the spin configuration in the high-field region, the M/H curves are investigated along the c-axis in Fig. 2f at the bias fields µ 0 H c = 1, 7, 8, and 9 T. A sharp drop near T N2 =2 Fig. 1d, pointing to the stabilization of the AFM 2 phase. When the bias field is increased, the M/H curves remain nearly unchanged up to 6 T, then increase conspicuously between µ 0 H c = 8 and 9 T, and becomes almost temperature-independent at H c = 9 T. This could result from an SF transition in the vicinity of µ 0 H c = 8 T. This observation is also consistent with the sharp increase of P a between µ 0 H c = 8 and 9 T. Such concomitant sharp increases of P a and M/H values imply that the SF phase of AFM 2 is responsible for the induced P a at µ 0 H c ≥ 6 T.
In order to check possible H-induced P a changes in other crystallographic directions and also at higher fields, we used a pulsed magnetic field up to 60 T to investigate the H-dependent ME current, magnetization (M), and ε curves in various H directions. Figure 3a summarizes the P(H) curve at 2 K for every H direction as obtained from the integration of the ME current. For H//c, P a becomes finite at~6 T and abruptly increases up toT able 1. Exchange parameters of PbCu 3 TeO 7 determined by the DFT calculation. J n (n = 1-8) refer to the intralayer exchange constants that appear in Fig. 1c, while J in (n = 1-4) represent the interlayer ones Cu-Cu distance, Å 6.353 6  15 µC/m 2 at~9 T; this is consistent with the P a data obtained from the DC magnetic fields from the I p measurement up to 14 T in Fig.  2. In this pulsed magnetic field experiment, we have verified that P a remains almost constant up to 60 T. Moreover, we found in the H//a (H a ) measurement that P a~1 4 µC/m 2 also develops at µ 0 H a ≥ 16 T and decreases again from µ 0 H a = 36 T to become zero at µ 0 H a = 41 T. Finally, we found no P a for H//b.  Fig. 3b, the fields where the SF transitions occur match those where the jumps in the P a (H) curves are observed. This implies that the changes in the spin ordering pattern induced by the magnetic fields should be responsible for the P a change. Furthermore, upon extrapolating the experimental magnetization data presented in Fig. 3b, we estimate the magnetic field for full saturation of magnetic moments to be~120-140 T.
To determine the phase boundaries of the SF transitions as functions of the temperature and magnetic field, we investigated the field-derivative curves of P a (dP a /dH) and M (dM/dH), as summarized in Fig. 3c, d. For H//c (Fig. 3c), all the dP a /dH, dM/dH, and ε a curves exhibit a single sharp peak at the SF transition; their peak fields are consistent with each other and become almost temperature-independent. Moreover, the peaks are observed only below 24 K, proving that the SF transition occurs in the AFM 2 phase. The peak fields become higher at higher temperatures and eventually disappear at 24 K, which agrees with the temperaturedependent measurements shown in Fig. 2. The evolution of the peak fields can be clearly seen in Fig. 4a, in which the electric and magnetic phase diagram is presented for H c .  However, for H//a, both dP a /dH and dM/dH curves display either peak or dip at µ 0 H a =~16 T and~38 T, pointing to the two consecutive SF transitions. The peak position at~16 T in the dP a / dH curve stays almost the same, while the second peak at~38 T shifts to lower fields upon warming up to 22 K. Furthermore, both peaks disappear above 24 K. The corresponding dM/dH curves exhibit a peak and a dip feature at almost same temperatures where the peaks in dP a /dH are observed. The peak and dip feature also disappears above 24 K. This directly proves that these fieldinduced SF transitions occur only within the AFM 2 phase stabilized below 24 K. The evolution of the peak or dip locations in dM/dH is also summarized in the phase diagram of Fig. 4b.
Based on the data presented in Figs. 2 and 3, we plot the phase diagram of PbCu 3 TeO 7 for H//c and H//a in Fig. 4a, b, respectively. The peak positions in dP a /dH, dM/dH, ε a (H), ε a (T), χ(H), and χ(T) and the temperatures, at which P a (T) starts to appear or disappear, are drawn. The phase boundaries from each measurement are consistent, displaying three major phases: a high-temperature paramagnetic paraelectric phase, the AFM 1 below T N1 = 36 K, and AFM 2 below T N2 = 24 K. In particular, we find evidence of finite electric polarization and ferroelectricity for H//c above 6 T with a concomitant, sharp SF transition. Thus, we represent the ferroelectric phase realized for H//c above 6 T as SF 1 and FE. Moreover, for H//a, we find two SF-like transitions inside the AFM 2 phase, and the resulting phases are termed SF 1' and SF 2. In particular, in the SF 1' region, we observe the ferroelectric polarization along the a-axis, as summarized in Fig. 4b.
In order to understand the possible magnetic structures in each phase and the H-induced electric polarization, we conducted classical MC simulations using the exchange constants, which were determined by the DFT calculation as described in the Methods section and are summarized in Table 1. The studied compound contains magnetic ions with lowest possible spin (S = 1/2) and, therefore, a variant of quantum MC simulations possibly would have been better suited to study its magnetic properties. However, we use here classical MC calculations to obtain a qualitative picture and use the results such as modulation wave vectors as inputs for the symmetry analysis of ME interactions described below. The results of MC simulations supported by the symmetry analysis are in good agreement with the experimental findings. The model Heisenberg Hamiltonian has the form, (1) where the spinsS i are classical vectors of unit length,H is the magnetic field, K α (α = x,y,z) are anisotropy constants, and we assume that the axes x, y, and z are along the a-, b-, and cdirections, respectively. According to the succession of anomalies in the χ (T) data, one can conclude that K x < K z < K y . We find that the simulation results based on the values K x = −1 meV, K y = 0.05 meV, and K z = 0 could provide qualitative correspondence with the experiments. Preliminary simulations of 20 × 20 × 20 unit cells (96,000 magnetic ions) first identified that the magnetic structure is modulated along the b-direction with a wave vector close tok ¼ 0; 1 3 ; 0 À Á . Therefore, subsequent studies were performed using the 4 × 75 × 4 simulation box.
In order to understand the magnetic structure and its coupling to the electric polarization, we performed a symmetry analysis. In thek ¼ 0; 1 3 ; 0 À Á point of the Brillouin-zone, the Pnma space group has four two-dimensional irreducible representations (IRs) Δ i (i = Fig. 5 Results of the MC simulation on PbCu 3 TeO 7 . a Temperature dependence of order parameter amplitudes A and C at a zero-applied magnetic field, b spatial variation of the order parameters (a1, a2) along the b-direction, and c the magnetization, d polarization, and e, f order parameters as functions of the magnetic field along various directions Magnetic field-induced ferroelectricity in S=1/2 kagomey K Yoo et al. 1,2,3,4). The magnetic structure belongs to the Δ 2 ⊗V′ exchange multiplet (here V′ is the pseudo-vector representation). 42 This multiplet splits into the direct sum of IRs Δ 4 , Δ 1 , and Δ 3 , according to which the x, y, and z components of spins transform, respectively, and we denote the respective order parameters by (a 1 ,a 2 ), (b 1 ,b 2 ), and (c 1 ,c 2 ).
The appearance of the modulated magnetic structure is in accordance with macroscopic symmetry since it allows the Lifshitz invariants which result in the modulation of magnetic order along the b-axis.
According to the MC simulations at a zero magnetic field, upon lowering the temperature, the first phase transition occurs at T MC N1 = 35 K and shows a spatially modulated order parameter (a 1 ,a 2 ). Further cooling results in the additional appearance of a spatially modulated (c 1 ,c 2 ) at T MC N2 = 28 K. Thus, the AFM 1 phase is sinusoidally modulated, whereas in the AFM 2 phase, spins lie in the ac-plane, forming a proper screw-type spin order. In the AFM 2 phase, the spin modulation period is approximately 25 unit cells along b. In spin-ordered phases AFM 1 and AFM 2, the spatially varying order parameters can be approximated with good accuracy by harmonic functions, e.g., a 1 = A Cos(k y+y 0 ) and a 2 = A sin(k y + y 0 ). Therefore, in the following, we define amplitudes A, B, and C of the order parameters. The temperature dependence of amplitudes A and C is shown in Fig. 5a, whereas an example of a spatially modulated order parameter is presented in Fig. 5b.
The dependence of the magnetization on external H applied along various directions is shown in Fig. 5c. Upon application of H c above ≈ 1.2 A.U., the plane of spin rotations changes into the abplane and becomes perpendicular to the magnetic field as shown in Fig. 5e. This cycloidal spin structure is described by spatially modulated (a 1 ,a 2 ) and (b 1 ,b 2 ) and possesses electric polarization along a due to the ME interaction in accordance with the experimental results for the SF 1 phase. However, the application of H a results in a more complex behavior. The symmetry allows the interaction where M a is the macroscopic magnetization along the a-axis. According to interaction (4), in the presence of the order parameters (a 1 ,a 2 ) and (c 1 ,c 2 ), the application of H a induces a spatially varying order parameter that is modulated with a wave vector twice that of (a 1 ,a 2 ) and (c 1 ,c 2 ) and which we denote by b 0 1 ; b 0 2 À Á to distinguish it from (b 1 ,b 2 ) modulated with the same k. The order parameters (b 1 ,b 2 ) and b 0 1 ; b 0 2 À Á have the same symmetry; however, according to the results of the MC calculations, the latter describes the magnetic moment components along the c-axis and is given by thus being proportional to H a . Here, we have b 0 The field dependences of A, B, B 0 , and C are shown in Fig. 5f. Due to the fact that b 0 1 ; b 0 2 À Á is modulated with a doubled wave vector, its appearance with the application of H a does not result in the emergence of electric polarization because the ME interaction represented by Eq. 3 averages to zero after the integration over the volume of the crystal. Therefore, model Hamiltonian (1) alone cannot explain the observation of P a in the SF 1 phase realized under intermediated H a .
However, our MC simulations show that further increases in H a above ≈ 5.3 A.U. eventually results in an SF transition, at which the plane of rotation changes to the bc-plane, as shown in Fig. 5f, being consistent with the experiments identifying the SF 2 phase. At intermediate H a < 5.3 A.U., it is possible to observe another field-induced phase transition, which can result either in a phase with all order parameters (a 1 ,a 2 ), b 0 1 ; b 0 2 À Á , and (c 1 ,c 2 ) being commensurate and non-zero or in an incommensurate phase with order parameters (a 1 ,a 2 ) and b 0 1 ; b 0 2 À Á modulated with the wave vectork and (c 1 ,c 2 ) with 2k. Both possible phases will result in the emergence of P a owing to Eq. 3. Such a field-induced phase transition can result from more complex magnetic anisotropies present in PbCu 3 TeO 7 , which are not captured in our simplified Hamiltonian. Thus, it represents a transition to a phase intermediate between the zero-field, ac proper screw, and high-field, bc cycloidal structures. Neutron diffraction studies are required in order to clarify the spin structure of this intermediate fieldinduced phase.
All of the spin structures obtained by the MC simulations are summarized in Fig. 6. In the ferroelectric phase, the spin structure is cycloidal and the spontaneous P emerges along the a-axis. The microscopic origin of P a can be revealed by rewriting Eq. 3 in Fig. 6 Schematics of spin arrangements in the a AFM 1, b AFM 2, c SF 1' , and d SF 1 and SF 2 phases in PbCu 3 TeO 7 . All the spin components were drawn based on the results of the MC simulations Magnetic field-induced ferroelectricity in S=1/2 kagomey K Yoo et al.
terms of spins. The corresponding analysis shows that P a arises due to contributions from both single-spin, described, e.g., by p-d hybridization model, and two-spin interactions, possessing vector spin products. The latter can be interpreted as IDM 22 or spin current 24 models. In the spin current model, the local electric polarization is expressed by where e ij is the propagation vector between S i and S j . In PbCu 3 TeO 7 , e ij is along the b-direction and the spin current j s ( = S i × S j ) is along the c-direction. Thus, the spin current model predicts the direction of P ij ( = e ij × j s ) to be along the a-axis, consistent with the experimental results. In addition, the polarizations P ij between each of the two spins have the same direction in the cycloidal order, which thus results in the macroscopic electric polarization P a .
In the SF 2 phase realized under H//a higher than 38 T, the spin rotation plane flips to the bc-plane. However, as the magnetic field is high, spins do not lie only in the bc-plane, but also are canted to the a-direction, resulting in the field-induced transverse conical (TC) spin order. In principle, the TC spin order should give rise to electric polarization P along the c-direction according to the spin current model. Macroscopic symmetry also allows a form of the ME interaction The MC simulations show that at H a > 5.3 A.U., the modulated order parameters (b 1 ,b 2 ) and (c 1 ,c 2 ) develop in a way that homogenous P c arises owing to Eq. 8. Thus, this TC phase is ferroelectric with the polarization along the c-direction. In the experiments, however, we have not confirmed P//c yet as the pulsed field experiments have been so far performed only at the P//a configuration.

DISCUSSION
The geometric spin frustration inherent to the kagome lattice generically suppresses the static long-range spin order. Therefore, ME/multiferroic properties, which are mostly coupled to a specific spin ordering, are rare in the kagome structure. However, the "distorted" kagome structures including the kagome staircase can relieve the degree of spin frustration either by anisotropic distortion in the kagome intralayer or by developing an interlayer interaction. As a result, several multiferroic/magnetoelectric compounds with the "distorted" kagome structures is characterized by only moderate spin frustration; for example, the ratio between the Curie-Weiss temperature and the Neel temperature θ CW /T N is~2 in KCu 3 As 2 O 7 (OD) 3 ,~3.3 in Ni 3 V 2 O 8 , and~4 in PbCu 3 TeO 7 . More importantly, such "distorted" kagome materials possess significant competing next-nearest interaction, which can often produce noncollinear spin orderings that are directly linked to their ferroelectric and ME properties. Therefore, significant exchange interactions beyond the first neighbors and the lifted geometric spin frustration should be regarded as common key features relevant for both Ni 3 V 2 O 8 and the presently studied PbCu 3 TeO 7 .
In recently reported multiferroic KCu 3 As 2 O 7 (OD) 3 37,38 with monoclinic two-dimensional kagome planes, the anisotropic intralayer distortion plays a dominant role in stabilizing a noncollinear spin ordering; it has nearly perfect two-dimensional kagome planes with the ratio of intralayer to interlayer interaction J inter /J intra~0 .01. 37 However, its intralayer distortion is significant enough to give rise to a high ratio of the intralayer lattice constants a/b of~1.72, thus relieving the geometric frustration in the kagome plane considerably and resulting in a long-range magnetic ordering below 7 K.
However, kagome staircase compound PbCu 3 TeO 7 studied here exhibits a much higher J inter /J intra~1 (Table 1), thereby more strongly mitigating spin frustration through the interlayer coupling. Owing to the inherent characteristic of the kagome staircase structure, the ratio of the intralayer lattice constants c/b of~1.4 in PbCu 3 TeO 7 is still high enough to cause further reduction in the degree of spin frustration. The situation for kagome staircase compound Ni 3 V 2 O 8 is similar; J inter /J intra~0 .4 9 and the ratio of the intralayer lattice constants is~1.39 13 . Therefore, both PbCu 3 TeO 7 and Ni 3 V 2 O 8 have a high number of considerable magnetic exchange interactions, which act together to reduce the geometric spin frustration and result in long-range incommensurately modulated magnetic ordering. External magnetic field perturbs the three-dimensional magnetic interactions in these staircase kagome compounds, which results in complex phase diagrams together with rich anisotropic behavior, as found in PbCu 3 TeO 7 as well as in Ni 3 V 2 O 8 .
The phase diagram of PbCu 3 TeO 7 can be compared with that of Ni 3 V 2 O 8 , where S = 1 and a cycloidal spin ordering is stabilized between 4 and 6.3 K in the low-temperature insulating phase. As another staircase kagome compound with S = 1/2, PbCu 3 TeO 7 similarly exhibits a kind of noncollinear spin ordering, i.e., a proper screw-type spin ordering, below T N2 = 24 K at a zero magnetic field. Both staircase compounds also form a collinear sinusoidal spin ordering at high temperatures above their noncollinear spin orderings. In this sense, they share close similarities in the spin ordering patterns. Furthermore, both compounds share another similarity of displaying complex electric/magnetic phase diagrams that are considerably different for each H direction.
However, Ni 3 V 2 O 8 has additional magnetic anisotropy, so that its spin rotation planes contain the spin propagation direction parallel to the Ni spins, forming a cycloid spin structure. Moreover, it exhibits two kinds of canted AFM order below 4 K, whereas PbCu 3 TeO 7 maintains a proper screw-type spin ordering below T N2 , whose spin rotation plane is perpendicular to the spin propagation direction parallel to Cu spines. Therefore, in PbCu 3 TeO 7 , the magnetic field is used to control effective magnetic anisotropy to further induce SF transitions and resultant spin cycloid phases with ferroelectric polarization. This fundamental difference in their magnetic anisotropy may stem from the fact that Ni 3 V 2 O 8 possesses a larger spin value (S = 1) than PbCu 3 TeO 7 (S = 1/2), thereby exhibiting additional sources of magnetic anisotropy due to zero-field splitting and stronger spin-orbit interactions. This may also explain why the polarization is larger in Ni 3 V 2 O 8 (~60 µC/m 2 ) 14 than PbCu 3 TeO 7 (~15 µC/m 2 ) and why the resultant field vs. temperature phase diagram of Ni 3 V 2 O 8 looks more complex than that of PbCu 3 TeO 7 .
Finally, we re-emphasize that the kagome staircase structure with both intralayer distortion and interlayer competing interaction can be a useful platform for searching for new field-induced ferroelectric polarization. In particular, the delicate balance in the magnetic anisotropy in three-dimensionally interacting spin networks can produce rich phase evolutions with various spin structures per different magnetic field directions. Therefore, our observation of field direction-dependent multiferroicity and nontrivial spin patterns in PbCu 3 TeO 7 , combined with former studies on Ni 3 V 2 O 8 , points to an opportunity that ample fieldinduced phases and ME/multiferroic effects can also be found in other staircase kagome structures.
In conclusion, we have investigated the ME properties in new staircase kagome magnet PbCu 3 TeO 7 with S = 1/2 Cu 2+ spins. The investigation of electric and magnetic properties has revealed that PbCu 3 TeO 7 becomes ferroelectric, exhibiting P a at µ 0 H c > 8.3 T and 15 T < µ 0 H a < 38 T and its field-induced phase transitions are concomitant with the SF transitions. By means of MC calculations, we establish a model of spin structures for each phase that mostly explains the origin of magnetic-field-induced P. Our investigation reveals that staircase kagome magnet PbCu 3 TeO 7 based on S = 1/ 2 (Cu 2+ ) is a fertile source of competing interactions that create ferroelectric polarization owing to the realization of various Magnetic field-induced ferroelectricity in S=1/2 kagomey K Yoo et al. inhomogeneous spin textures induced by magnetic fields. Further studies of neutron scattering and X-ray diffraction under a highmagnetic field will be helpful to fully identify the spin structures and to understand the origin of ME coupling in each high-field phase.

METHODS
Single crystals of PbCu 3 TeO 7 were grown using the flux method. First, a PbCu 3 TeO 7 polycrystalline sample was prepared by the solid-state reaction method using starting materials PbO, CuO, and Te. They were ground with a molar ratio of 1: 3: 1 and synthesized at 750°C for 5 days with three intermediate grindings. In order to grow single crystals, the polycrystalline of PbCu 3 TeO 7 and NaCl/KCl flux was mixed with a molar ratio of PbCu 3 TeO 7 : NaCl: KCl equal to 1: 10: 10. The mixture was heated at 800°C for 24 h and then cooled to 650°C at a rate of 1°C/h. The single crystals were extracted by washing with hot water.
For the electric measurements, we shaped the single crystal into thin plates, whose surface was normal to the crystal a-axis. The shaped crystal size was quite small with an area of 0.13 mm 2 and a thickness of 140 µm. The electrodes on both sides of the plate were formed with silver epoxy. We measured the pyroelectric and ME current (∝dP/dt) using KE617 electrometer and the dielectric constant using AH2550A capacitance bridge at a frequency of 1 kHz. The magnetic properties were collected using a vibrating sample magnetometer (VSM). These measurements were done in the 9 T PPMS TM (Quantum Design) and the 14 T Cryogenic LTD magnet system in the temperature (T) range from 2 K to 50 K.
The high-magnetic field ME current and magnetization measurements, using SR 570 current amplifier and the induction magnetometry technique, respectively, were done in the pulsed magnetic field up to 60 T at the NHMFL pulsed-field facility at LANL. The magnetization values from the pulsed field were calibrated using the VSM data measured in the 9 T PPMS TM .
The DFT calculations of the exchange constants in PbCu 3 TeO 7 were performed using the Vienna Ab initio Simulation Package. 43 We used the GGA+U approach with the on-site repulsion for Cu atoms by the Dudarev method. 44 The value U eff = U−J = 10 eV was chosen similar to other DFT studies of Cu-containing magneloelectrics. 45,46 The plane-wave cutoff energy of 500 eV was used, whereas the Brillouin-zone was integrated with the 4 × 6 × 4 mesh determined by the Monkhorst-Pack scheme. 47 In the previous work, 39 the most important exchange paths were studied identifying eight (J 1 to J 8 ) intra and four (J i1 to J i4 ) interkagome plane couplings. In the present work, we calculate the respective exchange constants, which are summarized in Table 1.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.